Plane-strain consolidation theory with distributed drainage boundary

Abstract

In practice, the full arrangement of sand blankets overlying soft clays could result in an uneconomic design for soft soil treatment using the surcharge preloading method. In view of this, a novel type of distributed drainage boundary is proposed in this investigation to improve the design. A two-dimensional plane-strain consolidation problem with distributed drainage boundary is established and solved. The sensitivity of the consolidation process to the pave rate (sand blanket area over the total area), thickness–width ratio (thickness over width of the representative element) and anisotropy coefficient (horizontal consolidation coefficient over vertical consolidation coefficient) are discussed. The results show that the negative effects of distributed drainage on the consolidation process become negligible if the pave rate and thickness–width ratio are designed adequately. Remarkably, the distributed drainage boundary becomes more efficient with the decrease in anisotropy coefficient. In addition, the increasing rate of consolidation time calculated using different parameters is presented for four average degrees of consolidation to provide a theoretical reference for engineering design.

Appendix

Applying the Laplace transform with respect to time factor T_v, the governing Eq. (12) is now written as an ordinary differential equation.

$$\frac{{\partial^{2} \bar{u}_{{\rm D}} }}{{\partial Z^{2} }} + \kappa \eta^{2} \frac{{\partial^{2} \bar{u}_{{\rm D}} }}{{\partial X^{2} }} - p\bar{u}_{{\rm D}} + 1 = 0,$$

(42)

with lateral boundary conditions

$$\left. {\frac{{\partial \bar{u}_{{\rm D}} }}{\partial X}} \right|_{X = 0} = \left. {\frac{{\partial \bar{u}_{{\rm D}} }}{\partial X}} \right|_{X = 1} = 0,$$

(43)

and vertical boundary conditions

$$\left. {\frac{{\partial \bar{u}_{{\rm D}} }}{\partial Z}} \right|_{Z = 0} = \left\{ {\begin{array}{*{20}l} {\bar{q}\left( {X,p} \right),} \hfill & {0 \le X \le \lambda } \hfill \\ {0,} \hfill & {\lambda < X \le 1} \hfill \\ \end{array} } \right.,$$

(44)

and

$$\left. {\frac{{\partial \bar{u}_{{\rm D}} }}{\partial Z}} \right|_{Z = 1} = 0,$$

(45)

where

$$\bar{u}_{{\rm D}} \left( {X,Z,p} \right) = \int\limits_{0}^{\infty } {u_{{\rm D}} \left( {X,Z,T_{{\rm v}} } \right){\text{e}}^{{ - pT_{{\rm v}} }} {\text{d}}T_{{\rm v}} };$$

p = Laplace transform variable; and $\bar{q}\left( {X,p} \right) =$ dimensionless drainage velocity in the Laplace domain.

According to the boundary conditions of Eq. (43), applying the finite Fourier cosine transform with respect to variable X, Eqs. (42), (44) and (45) can be expressed as

$$\frac{{\partial^{2} \tilde{\bar{u}}_{{\rm D}} }}{{\partial Z^{2} }} - \mu_{m}^{2} \tilde{\bar{u}}_{{\rm D}} + \delta_{m} = 0,$$

(46)

and

$$\left\{ \begin{array}{l} \left. {\frac{{\partial \tilde{\bar{u}}_{{\rm D}} }}{\partial Z}} \right|_{Z = 0} = \int_{0}^{\lambda } {\bar{q}\left( {X,p} \right){ \cos }\left( {M_{m} X} \right){\text{d}}X} \\ \left. {\frac{{\partial \tilde{\bar{u}}_{{\rm D}} }}{\partial Z}} \right|_{Z = 1} = 0 \\ \end{array} \right.,$$

(47)

where

$$\begin{aligned}& \tilde{\bar{u}}_{{\rm D}} \left( {m,Z,p} \right) = \int\limits_{0}^{1} {\bar{u}_{{\rm D}} \left( {X,Z,p} \right){ \cos }\left( {M_{m} X} \right){\text{d}}X}; \\ & \mu_{m} \left( p \right) = \sqrt {\kappa \eta^{2} M_{m}^{2} + p}; \ \delta_{m} = \left\{ {\begin{array}{*{20}c} {1,} & {m = 0} \\ {0,} & {m \ne 0} \\ \end{array} } \right.;\ M_m = m\pi;\end{aligned}$$

and m = Fourier transform variable.

The general solution of Eq. (46) with the boundary condition of Eq. (47) is derived as

$$\tilde{\bar{u}}_{{\rm D}} \left( {m,Z,p} \right) = \frac{{\delta_{m} }}{{\mu_{m}^{2} \left( p \right)}} - \frac{{\cosh \left( {\mu_{m} \left( p \right)\left( {1 - Z} \right)} \right)}}{{\mu_{m} \left( p \right)\sinh \left( {\mu_{m} \left( p \right)} \right)}}\times \int_{0}^{\lambda } {\bar{q}\left( {X,p} \right){ \cos }\left( {M_{m} X} \right){\text{d}}X}$$

(48)

By taking the inverse finite Fourier cosine transform to Eq. (48), the dimensionless pore-water pressure in the Laplace domain is obtained as

$$\bar{u}_{{\rm D}} \left( {X,Z,p} \right) = \bar{u}_{{\rm D1}} \left( {Z,p} \right) + \bar{u}_{{\rm D2}} \left( {X,Z,p} \right),$$

(49)

where

$$\bar{u}_{{\rm D1}} \left( {Z,p} \right) = \frac{1}{p} - \frac{{\cosh \left( {\mu_{0} \left( p \right)\left( {1 - Z} \right)} \right)}}{{\mu_{0} \left( p \right)\sinh \left( {\mu_{0} \left( p \right)} \right)}}\int_{0}^{\lambda } {\bar{q}\left( {X,p} \right){\text{d}}X} ,$$

and

$$\bar{u}_{{\rm D2}} \left( {X,Z,p} \right) = - 2\sum\limits_{m = 1}^{\infty } {\left[ {\frac{{{ \cos }\left( {M_{m} X} \right)\cosh \left( {\mu_{m} \left( p \right)\left( {1 - Z} \right)} \right)}}{{\mu_{m} \left( p \right)\sinh \left( {\mu_{m} \left( p \right)} \right)}}\times \int_{0}^{\lambda } {\bar{q}\left( {X,p} \right){ \cos }\left( {M_{m} X} \right){\text{d}}X} } \right]} .$$

The term $\bar{u}_{{\rm D1}} \left( {Z,p} \right)$ is independent of X, reflecting the horizontal averaged excess pore-water pressure. The integration of cos (M_mX) from 0 to 1 in $\bar{u}_{{\rm D2}}$ is zero, such that $\bar{u}_{{\rm D2}}$ can represent the distributed drainage effect.

Obviously, the unknown function $\bar{q}\left( {X,p} \right)$ becomes important for the derivation of solutions for Eq. (49). It can be obtained using the discretization method. The sand blanket length can be discretized into J segments with an element size of ΔX_j, and the corresponding dimensionless drainage velocity in the Laplace domain for segment j, $\bar{q}_{j} \left( p \right)$, can be assumed to be uniform (Fig. 16). In order to calculate $\bar{q}_{i} \left( p \right)$, the solution $\bar{u}_{{\rm D}}$ should satisfy the original drainage boundary in the Laplace domain. Corresponding to the discretized sand blanket, for any i ∊ [1, J], $\bar{u}_{{\rm D}}$ at the center of segment i should satisfy $\bar{u}_{{\rm D}} = 0$, that is

$$\sum\limits_{j = 1}^{J} {\left[ {\frac{{\coth \left( {\mu_{0} \left( p \right)} \right)}}{{\mu_{0} \left( p \right)}}\Delta X_{j} + 2\sum\limits_{m = 1}^{\infty } {\frac{{\coth \left( {\mu_{m} \left( p \right)} \right)}}{{\mu_{m} \left( p \right)}}g_{mj} \left( {X_{i} } \right)} } \right]\bar{q}_{j} \left( p \right)} = \frac{1}{p},$$

(50)

where $g_{mj} \left( X \right) = \frac{2}{{M_{m} }}\sin \left( {M_{m} \frac{{\Delta X_{j} }}{2}} \right)\cos \left( {M_{m} X_{j} } \right){ \cos }\left( {M_{m} X} \right)$; and X_j = the center coordinate of the jth segment.

Fig. 16

Discretization of sand blanket

For computational convenience, Eq. (50) can be rewritten in the matrix form as follows:

$$\sum\limits_{j = 1}^{J} {A_{ij} \left( p \right)\bar{q}_{j} \left( p \right)} = \frac{1}{p},\quad i = 1,2, \cdots ,J,$$

(51)

where $A_{ij} \left( p \right) = \frac{{\coth \left( {\mu_{0} \left( p \right)} \right)}}{{\mu_{0} \left( p \right)}}\Delta X_{i} + 2\sum\limits_{m = 1}^{\infty } {\frac{{\coth \left( {\mu_{m} \left( p \right)} \right)}}{{\mu_{m} \left( p \right)}}g_{mi} \left( {X_{j} } \right)}$.

From the expression of Eq. (51), one can obtain

$$\bar{q}_{j} \left( p \right) = \frac{1}{p}\sum\limits_{l = 1}^{J} {B_{jl} \left( p \right)} ,\quad j = 1,2, \cdots ,J,$$

(52)

where B_jl(p) = the elements of the matrix inversion of [A_ij(p)].

In the calculation of $\bar{q}_{j} \left( p \right)$, the convergence speed of A_ij determines the speed of the whole numerical analysis. In order to increase the calculation efficiency for series summation in A_ij, the infinite series can be effectively computed with the aid of the first-order Shanks transform [20]. Generally, the convergence of series summation in A_ij requires nearly 100 terms, depending on the discretization of the permeable boundary. In this study, the discretization is uniform and J is chosen around 10. Accordingly, after $\bar{q}_{j} \left( p \right)$ is obtained, the dimensionless pore-water pressure in the Laplace domain is obtained.